2,175 research outputs found
A path-integral approach to Bayesian inference for inverse problems using the semiclassical approximation
We demonstrate how path integrals often used in problems of theoretical
physics can be adapted to provide a machinery for performing Bayesian inference
in function spaces. Such inference comes about naturally in the study of
inverse problems of recovering continuous (infinite dimensional) coefficient
functions from ordinary or partial differential equations (ODE, PDE), a problem
which is typically ill-posed. Regularization of these problems using
function spaces (Tikhonov regularization) is equivalent to Bayesian
probabilistic inference, using a Gaussian prior. The Bayesian interpretation of
inverse problem regularization is useful since it allows one to quantify and
characterize error and degree of precision in the solution of inverse problems,
as well as examine assumptions made in solving the problem -- namely whether
the subjective choice of regularization is compatible with prior knowledge.
Using path-integral formalism, Bayesian inference can be explored through
various perturbative techniques, such as the semiclassical approximation, which
we use in this manuscript. Perturbative path-integral approaches, while
offering alternatives to computational approaches like Markov-Chain-Monte-Carlo
(MCMC), also provide natural starting points for MCMC methods that can be used
to refine approximations.
In this manuscript, we illustrate a path-integral formulation for inverse
problems and demonstrate it on an inverse problem in membrane biophysics as
well as inverse problems in potential theories involving the Poisson equation.Comment: Fixed some spelling errors and the author affiliations. This is the
version accepted for publication by J Stat Phy
Subsampling MCMC - An introduction for the survey statistician
The rapid development of computing power and efficient Markov Chain Monte
Carlo (MCMC) simulation algorithms have revolutionized Bayesian statistics,
making it a highly practical inference method in applied work. However, MCMC
algorithms tend to be computationally demanding, and are particularly slow for
large datasets. Data subsampling has recently been suggested as a way to make
MCMC methods scalable on massively large data, utilizing efficient sampling
schemes and estimators from the survey sampling literature. These developments
tend to be unknown by many survey statisticians who traditionally work with
non-Bayesian methods, and rarely use MCMC. Our article explains the idea of
data subsampling in MCMC by reviewing one strand of work, Subsampling MCMC, a
so called pseudo-marginal MCMC approach to speeding up MCMC through data
subsampling. The review is written for a survey statistician without previous
knowledge of MCMC methods since our aim is to motivate survey sampling experts
to contribute to the growing Subsampling MCMC literature.Comment: Accepted for publication in Sankhya A. Previous uploaded version
contained a bug in generating the figures and reference
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